Properties

Label 2240.2.b.c
Level $2240$
Weight $2$
Character orbit 2240.b
Analytic conductor $17.886$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{3} q^{5} - q^{7} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{12}^{2} ) q^{3} -\zeta_{12}^{3} q^{5} - q^{7} -\zeta_{12}^{3} q^{11} -\zeta_{12}^{3} q^{13} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{15} + ( -4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{17} -2 \zeta_{12}^{3} q^{19} + ( -1 + 2 \zeta_{12}^{2} ) q^{21} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{23} - q^{25} + ( 3 - 6 \zeta_{12}^{2} ) q^{27} + ( -3 + 6 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{29} -4 q^{31} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{33} + \zeta_{12}^{3} q^{35} + ( -2 + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{37} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{39} + ( -4 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{41} + ( 2 - 4 \zeta_{12}^{2} ) q^{43} + ( -3 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{47} + q^{49} + ( -4 + 8 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{51} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{53} - q^{55} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( 2 - 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{59} + ( -2 + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{61} - q^{65} + ( -2 + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{67} + 6 \zeta_{12}^{3} q^{69} + ( -4 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{71} + 4 q^{73} + ( -1 + 2 \zeta_{12}^{2} ) q^{75} + \zeta_{12}^{3} q^{77} + ( -4 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{79} -9 q^{81} + ( -2 + 4 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{83} + ( 1 - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{85} + ( 9 - 8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{87} + ( 6 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{89} + \zeta_{12}^{3} q^{91} + ( -4 + 8 \zeta_{12}^{2} ) q^{93} -2 q^{95} + ( -16 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} - 16q^{17} - 4q^{25} - 16q^{31} - 16q^{41} - 12q^{47} + 4q^{49} - 4q^{55} - 4q^{65} - 16q^{71} + 16q^{73} - 16q^{79} - 36q^{81} + 36q^{87} + 24q^{89} - 8q^{95} - 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1121.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 1.73205i 0 1.00000i 0 −1.00000 0 0 0
1121.2 0 1.73205i 0 1.00000i 0 −1.00000 0 0 0
1121.3 0 1.73205i 0 1.00000i 0 −1.00000 0 0 0
1121.4 0 1.73205i 0 1.00000i 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.b.c 4
4.b odd 2 1 2240.2.b.d yes 4
8.b even 2 1 inner 2240.2.b.c 4
8.d odd 2 1 2240.2.b.d yes 4
16.e even 4 1 8960.2.a.z 2
16.e even 4 1 8960.2.a.bb 2
16.f odd 4 1 8960.2.a.y 2
16.f odd 4 1 8960.2.a.ba 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.b.c 4 1.a even 1 1 trivial
2240.2.b.c 4 8.b even 2 1 inner
2240.2.b.d yes 4 4.b odd 2 1
2240.2.b.d yes 4 8.d odd 2 1
8960.2.a.y 2 16.f odd 4 1
8960.2.a.z 2 16.e even 4 1
8960.2.a.ba 2 16.f odd 4 1
8960.2.a.bb 2 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{2} + 3 \)
\( T_{23}^{2} - 12 \)
\( T_{31} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( ( 1 + T^{2} )^{2} \)
$13$ \( ( 1 + T^{2} )^{2} \)
$17$ \( ( 13 + 8 T + T^{2} )^{2} \)
$19$ \( ( 4 + T^{2} )^{2} \)
$23$ \( ( -12 + T^{2} )^{2} \)
$29$ \( 121 + 86 T^{2} + T^{4} \)
$31$ \( ( 4 + T )^{4} \)
$37$ \( 16 + 56 T^{2} + T^{4} \)
$41$ \( ( -32 + 8 T + T^{2} )^{2} \)
$43$ \( ( 12 + T^{2} )^{2} \)
$47$ \( ( -39 + 6 T + T^{2} )^{2} \)
$53$ \( 64 + 32 T^{2} + T^{4} \)
$59$ \( 576 + 96 T^{2} + T^{4} \)
$61$ \( 64 + 32 T^{2} + T^{4} \)
$67$ \( 64 + 32 T^{2} + T^{4} \)
$71$ \( ( 4 + 8 T + T^{2} )^{2} \)
$73$ \( ( -4 + T )^{4} \)
$79$ \( ( 13 + 8 T + T^{2} )^{2} \)
$83$ \( 17424 + 312 T^{2} + T^{4} \)
$89$ \( ( 24 - 12 T + T^{2} )^{2} \)
$97$ \( ( 253 + 32 T + T^{2} )^{2} \)
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